Naturality of support for tilting modules under Levi restriction

Let $\mathfrak{g}$ be a Lie algebra satisfying the assumptions of Section, let $\mathfrak{l}\subset\mathfrak{g}$ be a Levi subalgebra, and let $\operatorname{Res}$ denote restriction from tilting $U_\zeta(\mathfrak{g})$-modules to tilting $U_\zeta(\mathfrak{l})$-modules. For a tilting module $T$, write $\operatorname{supp}T$ for its support, and let $\mathscr{N}(\mathfrak{l})$ denote the relevant nilpotent cone. The conditions referred to are the equivalent support and ideal conditions in Proposition~.

Naturality conjecture. The conditions in Proposition~ are satisfied for an arbitrary $\mathfrak{g}$, with the assumptions of Section; equivalently, support of tilting modules is natural with respect to Levi subalgebras.

The type-$\mathsf{A}$ case is proved in the paper. The conjecture is motivated by the expectation that the relevant support inclusion holds even for arbitrary modules, but the general case remains open.